The fact that \(u_\varepsilon \)  verifies (REF ) is a consequence of the fact that
any solution of (REF ) with the specified regularity satisfies the following energy estimate.
\(\frac{1}{\vert \log \varepsilon \vert }\int _{0}^T\int _{M}\vert \partial _t u_\varepsilon \vert ^2_g \, \mathrm {vol}_g\mathrm {d}t + F_\varepsilon (u_\varepsilon (t)) = F_\varepsilon (u_0)\le \pi n\vert \log \varepsilon \vert + C,\) 
where \(C\)  denote a constant independent of \(\varepsilon \) .
Therefore (REF ) follows.
Set \(v_\varepsilon (x,t\vert \log \varepsilon \vert ) := u_\varepsilon (x,t)\)  for \((x,t)\in [0, \, T]\times M\) . Then, \(v_\varepsilon \)  solves
\({\left\lbrace \begin{array}{ll}\displaystyle \partial _t v_\varepsilon -\Delta _g v_\varepsilon + \displaystyle \frac{1}{\varepsilon ^2}(\vert v_\varepsilon \vert ^2_g-1)v_\varepsilon = 0,\,\,\,\,\hbox{ a.e. in }M\times (0,T)\\[.3cm]\displaystyle v_\varepsilon (x,0) = u_0\,\,\,\,\hbox{ a.e. in }M.\end{array}\right.}\) 
Then, test (in the scalar product \((\cdot , \cdot )_g\) ) with \(v_\varepsilon \) .
We get
\(\partial _t \vert v_\varepsilon \vert ^2_g -\Delta _g \vert v_\varepsilon \vert ^2_g + 2 \vert v̥_\varepsilon \vert ^2_g + \frac{2}{\varepsilon ^2}\vert v_\varepsilon \vert ^2_g (\vert v_\varepsilon \vert ^2_g -1) =0.\) 
Therefore, if we set \(w_\varepsilon :=\vert v_\varepsilon \vert ^2_g -1\) , we obtain that
\(\partial _t w_\varepsilon -\Delta _g w_\varepsilon + \frac{2}{\varepsilon ^2}w_\varepsilon \le 0,\) 
and thus \(w_\varepsilon (t)\le w_\varepsilon (0) \le 0\) , namely \(\vert v_\varepsilon \vert _g \le 1\) .
By rescaling back with respect to time we obtain (REF ).
Finally, we discuss the validity of (REF ). In particular, note that that it is sufficient to prove that there exists some \(T_0>0\)  such that
\(\int _{0}^{T_0}\int _{M}\vert \partial _t u_\varepsilon \vert ^2_g \, \mathrm {vol}_g\mathrm {d}t \le \vert \log \varepsilon \vert ,\) 
but this has been already proved in [1]}. The argument uses a contradiction argument combined with the product estimate (REF ) and can be replicated verbatim.
